On (p1,…,pk)-spherical distributions

Abstract

The class of (p 1 ,. .. , p k)-spherical probability laws and a method of simulating random vectors following such distributions are introduced using a new stochastic vector representation. A dynamic geometric disintegration method and a corresponding geometric measure representation are used for generalizing the classical χ 2-, t-and F-distributions. Comparing the principles of specialization and marginalization gives rise to an alternative method of dependence modeling.

Modelling with star-shaped distributions

“…However, on using the common multivariate polar or spherical coordinate transformation and its well known inverse, we can introduce another model class. M 7: For η = p > , In case that B = {x ∈ R d : x ≤ }, Model class 3 may directly be generalized to the multivariate case, meaning that equations (19) up to (22) are valid also for x ∈ R d with the norms correspondingly de ned there.…”

“…The Minkowski functional of K, h K (x) = inf{λ > : x ∈ λK}, is de ned for every K under consideration and may particularly be any norm or antinorm. Although, for mathematical reasons discussed for a particular case in [19], and more generally in [22], it is not trivial to further assume that the Minkowski functional h K of K is positively homogeneous of degree one. This restriction made here might be considered not to be too restrictive in many applied situations.…”

On Ball Numbers

“…is aimed to be a probability density, then it follows from [26] that the normalizing constant allows the representation C(g; p) = 1 nI(g; p)π(B p ) .…”

“…This particular density is called the p-spherical Pearson Type II density with parameter ν > 0. For two more particular cases, the p-spherical Pearson Type VII and Kotz type densities, we refer to [26]. Let us recall that there and in the present example p is a vector having different positive components.…”

“…dx can be evaluated by changing Cartesian with (p, ℘)-spherical coordinates in the sense of Definition 1 in [26] where the formal parameter is chosen as ℘ = 1. According to Equation (6) in [26], where π * p = π p , we have similarly to Theorem 3 in [19] that…”